PQRS is a ‖parallelogram . M is a point on PS such that PM = 7 PS and N is apoint on QR such that RN = 7 QR. Prove that the quadrilateral PNRM is a ‖gm.
Question
PQRS is a |
parallelogram. M is a point on PS such that PM = 7 PS and N is a point on QR such that RN = 7 QR. Prove that the quadrilateral PNRM is a |gm.
Solution
To prove that the quadrilateral PNRM is a parallelogram, we need to show that its opposite sides are parallel.
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Given that PQRS is a parallelogram, we know that its opposite sides are parallel. Therefore, PQ || SR and PS || QR.
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Let's consider the line segment PM. We are given that PM = 7PS. This means that PM is 7 times longer than PS.
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Similarly, let's consider the line segment RN. We are given that RN = 7QR. This means that RN is 7 times longer than QR.
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Now, let's draw a line segment from P to R. Since PQ || SR, the line segment PR is parallel to QR.
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Since PM is 7 times longer than PS and RN is 7 times longer than QR, it follows that PR is also 7 times longer than PS.
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Therefore, we can conclude that PR is parallel to PS.
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Similarly, we can draw a line segment from N to M. Since PS || QR, the line segment NM is parallel to PS.
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Since RN is 7 times longer than QR and PM is 7 times longer than PS, it follows that NM is also 7 times longer than PS.
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Therefore, we can conclude that NM is parallel to PS.
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We have shown that PR is parallel to PS and NM is parallel to PS. This means that PNRM is a quadrilateral with opposite sides that are parallel.
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Hence, we have proved that the quadrilateral PNRM is a parallelogram.
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